Machine Learning - MT Clustering

Transcription

1 Machine Learning - MT Clustering Varun Kanade University of Oxford November 28, 2016

2 Announcements No new practical this week All practicals must be signed off in sessions this week Firm Deadline: Reports handed in at CS reception by Friday noon Revision Class for M.Sc. + D.Phil. Thu Week 9 (2pm & 3pm) Work through ML HT2016 Exam (Problem 3 is optional) 1

3 Outline This week, we will study some approaches to clustering Defining an objective function for clustering k-means formulation for clustering Multidimensional Scaling Hierarchical clustering Spectral clustering 2

4 England pushed towards Test defeat by India France election: Socialists scramble to avoid split after Fillon win Giants Add to the Winless Browns Misery Strictly Come Dancing: Ed Balls leaves programme Trump Claims, With No Evidence, That Millions of People Voted Illegally Vive La Binoche, the reigning queen of French cinema 3

5 Sports Politics Sports Film&TV Politics Film&TV England pushed towards Test defeat by India France election: Socialists scramble to avoid split after Fillon win Giants Add to the Winless Browns Misery Strictly Come Dancing: Ed Balls leaves programme Trump Claims, With No Evidence, That Millions of People Voted Illegally Vive La Binoche, the reigning queen of French cinema 3

6 England France USA England USA France England pushed towards Test defeat by India France election: Socialists scramble to avoid split after Fillon win Giants Add to the Winless Browns Misery Strictly Come Dancing: Ed Balls leaves programme Trump Claims, With No Evidence, That Millions of People Voted Illegally Vive La Binoche, the reigning queen of French cinema 3

7 Clustering Often data can be grouped together into subsets that are coherent. However, this grouping may be subjective. It is hard to define a general framework. Two types of clustering algorithms 1. Feature-based - Points are represented as vectors in R D 2. (Dis)similarity-based - Only know pairwise (dis)similarities Two types of clustering methods 1. Flat - Partition the data into k clusters 2. Hierarchical - Organise data as clusters, clusters of clusters, and so on 4

8 Defining Dissimilarity Weighted dissimilarity between (real-valued) attributes D d(x, x ) = f w id i(x i, x i) i=1 In the simplest setting w i = 1 and d i(x i, x i) = (x i x i) 2 and f(z) = z, which corresponds to the squared Euclidean distance Weights allow us to emphasise features differently If features are ordinal or categorical then define distance suitably Standardisation (mean 0, variance 1) may or may not help 5

9 Helpful Standardisation 6

10 Unhelpful Standardisation 7

11 Partition Based Clustering Want to partition the data into subsets C 1,..., C k, where k is fixed in advance Define quality of a partition by W (C) = 1 2 k j=1 If we use d(x, x ) = x x 2, then where µ j = 1 C j i C j x i W (C) = 1 C j k j=1 i,i C j d(x i, x i ) i C j x i µ j 2 The objective is minimising the sum of squares of distances to the mean within each cluster 8

12 Outline Clustering Objective k-means Formulation of Clustering Multidimensional Scaling Hierarchical Clustering Spectral Clustering

13 Partition Based Clustering : k-means Objective Minimise jointly over partitions C 1,..., C k and µ 1,..., µ k W (C) = k j=1 i C j x i µ j 2 This problem is NP-hard even for k = 2 for points in R D If we fix µ 1,..., µ j, finding a partition (C j) k j=1 that minimises W is easy C j = {i x i µ j = min x i µ j } j If we fix the clusters C 1,..., C k minimising W with respect to (µ j ) k j=1 is easy µ j = 1 x i C j i C j Iteratively run these two steps - assignment and update 9

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19 Ground Truth Clusters k-means Clusters (k = 3) 11

20 The k-means Algorithm 1. Intialise means µ 1,..., µ k randomly 2. Repeat until convergence: a. Find assignments of data to clusters represented by the mean that is closest to obtain, C 1,..., C k : C j = {i j = argmin j x i µ j 2 } b. Update means using the current cluster assignments: µ j = 1 x i C j i C j Note 1: Ties can be broken arbitrarily Note 2: Choosing k random datapoints to be the initial k-means is a good idea 12

21 The k-means Algorithm Does the algorithm always converge? Yes, because the W function decreases every time a new partition is used; there are only finitely many partitions W (C) = k j=1 i C j x i µ j 2 Convergence may be very slow in the worst-case, but typically fast on real-world instances Convergence is probably to a local minimum. Run multiple times with random initialisation. Can use other criteria: k-medoids, k-centres, etc. Selecting the right k is not easy: plot W against k and identify a "kink" 13

22 Ground Truth Clusters k-means Clusters (k = 4) 14

23 Choosing the number of clusters k 0.25 MSE on test vs K for K means As in the case of PCA, larger k will give better value of the objective Choose suitable k by identifying a kink or elbow in the curve (Source: Kevin Murphy, Chap 11) 15

24 Outline Clustering Objective k-means Formulation of Clustering Multidimensional Scaling Hierarchical Clustering Spectral Clustering

25 Multidimensional Scaling (MDS) In certain cases, it may be easier to define (dis)similarity between objects than embed them in Euclidean space Algorithms such as k-means require points to be in Euclidean space Ideal Setting: Suppose for some N points in R D we are given all pairwise Euclidean distances in a matrix D Can we reconstruct x 1,..., x N, i.e., all of X? 16

26 Multidimensional Scaling Distances are preserved under translation, rotation, reflection, etc. We cannot recover X exactly; we can aim to determine X up to these transformations If D ij is the distance between points x i and x j, then D 2 ij = x i x j 2 = x T i x i 2x T i x j + x T j x j = M ii 2M ij + M jj Here M = XX T is the N N matrix of dot products Exercise: Show that assuming i xi = 0, M can be recovered from D 17

27 Multidimensional Scaling Consider the (full) SVD: X = UΣV T We can write M as M = XX T = UΣΣ T U T Starting from M, we can reconstruct X using the eigendecomposition of M M = UΛU T Because, M is symmetric and positive semi-definite, U T = U 1 and all entries of (diagonal matrix) Λ are non-negative Let X = UΛ 1/2 If we are satisfied with approximate reconstruction, we can use truncated eigendecomposition 18

28 Multidimensional Scaling: Additional Comments In general if you define (dis)similarities on objects such as text documents, genetic sequences, etc., we cannot be sure that the generated similarity matrix M will be positive semi-definite or that the dissimilarity matrix D is a valid squared Euclidean distance If such cases, we cannot always find a Euclidean embeddding that recovers the (dis)similarities exactly Minimize stress function: Find z 1,..., z N that minimizes S(Z) = i j (D ij z i z j ) 2 Several other types of stress functions can be used 19

29 Multidimensional Scaling: Summary In certain applications, it may be easier to define pairwise similarities or distances, rather than construct a Euclidean embedding of discrete objects, e.g., genetic data, text data, etc. Many machine learning algorithms require (or are more naturally expressed with) data in some Euclidean space Multidimensional Scaling gives a way to find an embedding of the data in Euclidean space that (approximately) respects the original distance/similarity values 20

30 Outline Clustering Objective k-means Formulation of Clustering Multidimensional Scaling Hierarchical Clustering Spectral Clustering

31 Hierarchical Clustering Hierarchical structured data exists all around us Measurements of different species and individuals within species Top-level and low-level categories in news articles Country, county, town level data Two Algorithmic Strategies for Clustering Agglomerative: Bottom-up, clusters formed by merging smaller clusters Divisive: Top-down, clusters formed by splitting larger clusters Visualise this as a dendogram or tree 21

32 Measuring Dissimilarity at Cluster Level To find hierarchical clusters we need to define dissimilarity at cluster level, not just at datapoints Suppose we have dissimilarity at datapoint level, e.g., d(x, x ) = x x Different ways to define dissimilarity at cluster level, say C and C Single Linkage Complete Linkage D(C, C ) = D(C, C ) = min d(x, x ) x C,x C max d(x, x ) x C,x C Average Linkage D(C, C ) = 1 C C x C,x C d(x, x ) 22

33 Measuring Dissimilarity at Cluster Level Single Linkage D(C, C ) = min d(x, x ) x C,x C Complete Linkage D(C, C ) = max d(x, x ) x C,x C Average Linkage D(C, C ) = 1 C C x C,x C d(x, x ) 23

34 Linkage-based Clustering Algorithm 1. Initialise clusters as singletons C i = {i} 2. Initialise clusters available for merging S = {1,..., N} 3. Repeat a. Pick 2 most similar clusters, (j, k) = argmin D(j, k) j,k S b. Let C l = C j C k c. If C l = {1,..., N}, break; d. Set S = (S \ {j, k}) {l} e. Update D(i, l) for all i S (using desired linkage property) 24

35 Hierarchical Clustering: Dendogram Outputs of hierarchical clustering algorithms are typically represented using dendograms A dendogram is a binary tree, representing clusters as they were merged The height of a node represents dissimilarity Cutting the dendogram at some level gives a partition of data 25

36 Outline Clustering Objective k-means Formulation of Clustering Multidimensional Scaling Hierarchical Clustering Spectral Clustering

37 Spectral Clustering 26

38 Spectral Clustering: Limitations of k-means 27

39 Limitations of k-means k-means will typically form clusters that are spherical, elliptical, convex Kernel PCA followed by k-means can result in better clusters Spectral clustering is a (related) alternative that often works better 28

40 Spectral Clustering Construct a graph from data; one node for every point in dataset Use similarity measure, e.g., s i,j = exp( x i x j 2 /σ) Construct mutual K-nearest neighbour graph, i.e., (i, j) is an edge if either i is among the K nearest neighbours of j or vice versa The weight of edge (i, j), if it exists is s i,j 29

41 Spectral Clustering 30

42 Spectral Clustering Use graph partitioning algorithms Mincut can give bad cuts (only one node on one side of the cut) Multi-way cuts, balanced cuts, are typically NP-hard to compute Relaxations of these problems give eigenvectors of Laplacian W is the weighted adjacency matrix D is (diagonal) degree matrix: D ii = j Wij Laplacian L = D W Normalised Laplacian: L = I D 1 W 31

43 Spectral Clustering: Simple Example Suppose all edge weights are 1 (0 for missing edges) The weighted adjacency matrix, the degree matrix and the Laplacian are given by W = D = L = D W =

44 Spectral Clustering: Simple Example Let us consider some eigenvectors of L L = D W = Suppose all edge weights are 1 (0 for missing edges) v 1 = [1, 1, 1, 1, 1, 1] T is an eigenvector with eigenvalue 0 v 2 = [1, 1, 1, 1, 1, 1] T is also an eigenvector with eigenvalue 0 α 1v 1 + α 2v 2 for any α 1, α 2 is also an eigenvector with eigenvalue 0 We can use the matrix [v 1v 2] as the N 2 feature matrix and perform k-means 33

45 Spectral Clustering: Simple Example Suppose all edge weights are 1 (0 for missing edges) 34

46 Spectral Clustering: Simple Example Suppose all edge weights are 1 (0 for missing edges) Let us consider some eigenvectors of L L = D W = When the weights are slightly perturbed, v 1 = [1,..., 1] T is still an eigenvector with eigenvalue 1 We can t compute the second eigenvector v 2 by hand Nevertheless, we expect that the eigenspace corresponding to similar eigenvalues is relatively stable We can still use the matrix [v 1v 2] as the N 2 feature matrix and perform k-means 35

47 Spectral Clustering: Simple Example Suppose all edge weights are 1 (0 for missing edges) 36

48 Spectral Clustering Algorithm Input: Weighted graph with weighted adjacency matrix W 1. Construct Laplacian L = D W 2. Find v 1 = 1, v 2,..., v l+1 the k-eigenvectors 3. Construct the N l feature matrix V l = [v 2,, v l ] 4. Apply clustering algorithm using V l as features, e.g., k-means Note: If the degrees of nodes are not balanced, using the normalised Laplacian, L = I D 1 W may be a better idea 37

49 Spectral Clustering 38

50 Summary: Clustering Clustering is grouping together similar data in a larger collection of heterogeneous data Definition of good clusters often user-dependent Clustering algorithms in feature space, e.g., k-means Clustering algorithms that only use (dis)similarities: k-medoids, hierarchical clustering Spectral clustering when clusters may be non-convex 39